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The Detweiler-Barack-Sago redshift function for particles moving alongslightly eccentric equatorial orbits around a Kerr black hole is currentlyknown up to the second order in eccentricity, second order in spin parameter,and the 8.5 post-Newtonian order. We improve the analytical computation of sucha gauge-invariant quantity by including terms up to the fourth order ineccentricity at the same post-Newtonian approximation level. We also check thatour results agrees with the corresponding post-Newtonian expectation of thesame quantity, calculated by using the currently known Hamiltonian for spinningbinaries.

New gravitational self-force analytical results for eccentric equatorial orbits around a Kerr black hole: redshift invariant